cdfpe3 {lmomco} | R Documentation

## Cumulative Distribution Function of the Pearson Type III Distribution

### Description

This function computes the cumulative probability or nonexceedance probability of the Pearson Type III distribution given parameters (μ, σ, and gamma) of the distribution computed by `parpe3`. These parameters are equal to the product moments: mean, standard deviation, and skew (see `pmoms`). The cumulative distribution function of the distribution for gamma ne 0 is

F(x) = frac{G(α,frac{Y}{β})}{Γ(α)} mbox{,}

where F(x) is the nonexceedance probability for quantile x, G is defined below and is related to the incomplete gamma function of R (`pgamma()`), Γ is the complete gamma function, xi is a location parameter, β is a scale parameter, α is a shape parameter, and Y = x - xi if gamma > 0 and Y = xi - x if gamma < 0. These three “new” parameters are related to the product moments by

α = 4/gamma^2 mbox{,}

β = frac{1}{2}σ |gamma| mbox{,}

xi = μ - 2σ/gamma mbox{.}

The function G(α,x) is

G(α,x) = int_0^x t^{(a-1)} mathrm{e}^{-t} mathrm{d}t mbox{.}

If gamma = 0, the distribution is symmetrical and simply is the normal distribution with mean and standard deviation of μ and σ, respectively. Internally, the gamma = 0 condition is implemented by `pnorm()`. If gamma > 0, the distribution is right-tail heavy, and F(x) is the returned nonexceedance probability. On the other hand if gamma < 0, the distribution is left-tail heavy and 1-F(x) is the actual nonexceedance probability that is returned.

### Usage

    cdfpe3(x, para)

### Arguments

`x` | A real value.
`para` | The parameters from `parpe3` or similar.

### Value

Nonexceedance probability (F) for x.

### Author(s)

W.H. Asquith

### References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

### See Also

`quape3`, `parpe3`

### Examples

      lmr <- lmom.ub(c(123,34,4,654,37,78))
      cdfpe3(50,parpe3(lmr))

[Package lmomco version 0.92 Index]